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<CENTER><FONT color=darkblue><B>Acoustics Animations</B></FONT><BR><FONT 
color=darkred><B>Dr. Dan Russell</B><BR>Kettering University Applied Physics 
</FONT></CENTER>
<HR>

<H1>Reflection of Waves from Boundaries</H1><FONT size=2>These animations were 
inspired in part by the figures in chapter 6 of <I>Introduction to Wave 
Phenomena</I> by A. Hirose and K. Lonngren, (J. Wiley &amp; Sons, 1985, 
reprinted by Kreiger Publishing Co., 1991)</FONT> 
<P>When an object, like a ball, is thrown against a rigid wall it bounces back. 
This "reflection" of the object can be analyzed in terms of momentum and energy 
conservation. If the collision between ball and wall is perfectly elastic, then 
all the incident energy and momentum is reflected, and the ball bounces back 
with the same speed. If the collision is inelastic, then the wall (or ball) 
absorbs some of the incident energy and momentum and the ball does not bounce 
back with the same speed. 
<P>Waves also carry energy and momentum, and whenever a wave encounters an 
obstacle, they are reflected by the obstacle. This reflection of waves is 
responsible for echoes, radar detectors, and for allowing <A 
href="http://www.kettering.edu/~drussell/Demos/superposition/superposition.html#standing">standing 
waves</A> which are so important to sound production in musical instruments. 
<HR>

<H2>Wave pulse travelling on a string </H2>
<TABLE>
  <TBODY>
  <TR>
    <TD><IMG src="Reflection of Waves from Boundaries_files/pulse.gif" 
      width=200 align=left></TD>
    <TD vAlign=top>The animation at left shows a wave pulse travelling on a 
      string. The speed, <I>c</I>, with which the wave pulse travels along the 
      string depends on the elastic restoring force (tension, <I>T</I>) and 
      inertia (mass per unit length, <IMG 
      src="Reflection of Waves from Boundaries_files/mu.gif">) according to <BR>
      <CENTER><IMG 
      src="Reflection of Waves from Boundaries_files/Eq-2.jpg"></CENTER><BR>.</TD></TR></TBODY></TABLE>
<HR>

<H2>Reflection from a HARD boundary</H2>
<TABLE>
  <TBODY>
  <TR>
    <TD><IMG src="Reflection of Waves from Boundaries_files/hard.gif" 
      width=300 align=left></TD>
    <TD vAlign=top>The animation at left shows a wave pulse on a string moving 
      from left to right towards the end which is rigidly clamped. As the wave 
      pulse approaches the fixed end, the internal restoring forces which allow 
      the wave to propagate exert an upward force on the end of the string. But, 
      since the end is clamped, it cannot move. According to Newton's third law, 
      the wall must be exerting an equal downward force on the end of the 
      string. This new force creates a wave pulse that propagates from right to 
      left, with the same speed and amplitude as the incident wave, but with 
      opposite polarity (upside down).<BR>
      <DL>
        <DT>=&gt; <I>at a fixed (hard) boundary, the displacement remains zero 
        and the reflected wave changes its polarity (undergoes a 180<SUP>o</SUP> 
        phase change)</I> </DT></DL></TD></TR></TBODY></TABLE>
<P>
<HR>

<H2>Reflection from a SOFT boundary</H2>
<TABLE>
  <TBODY>
  <TR>
    <TD><IMG src="Reflection of Waves from Boundaries_files/soft.gif" 
      width=300 align=left></TD>
    <TD vAlign=top>The animation at left shows a wave pulse on a string moving 
      from left to right towards the end which is free to move vertically 
      (<SMALL>imagine the string tied to a massless ring which slides 
      frictionlessly up and down a vertical pole</SMALL>). The net vertical 
      force at the free end must be zero. This boundary condition is 
      mathematically equivalent to requiring that the slope of the string 
      displacement be zero at the free end (look closely at the movie to verify 
      that this is true). The reflected wave pulse propagates from right to 
      left, with the same speed and amplitude as the incident wave, and with the 
      same polarity (rightside up).<BR>
      <DL>
        <DT>=&gt; <I>at a free (soft) boundary, the restoring force is zero and 
        the reflected wave has the same polarity (no phase change) as the 
        incident wave</I> </DT></DL></TD></TR></TBODY></TABLE>
<P>
<HR>

<H2>Reflection from an impedance discontinuity</H2>When a wave encounters a 
boundary which is neither rigid (hard) nor free (soft) but instead somewhere in 
between, part of the wave is reflected from the boundary and part of the wave is 
transmitted across the boundary. The exact behavior of reflection and 
transmission depends on the material properties on both sides of the boundary. 
One important property is the <I>characterstic impedance</I> of the material. 
The chracterstic impedance of a material is the product of mass density and wave 
speed, <IMG src="Reflection of Waves from Boundaries_files/zrhoc.gif">. If a 
wave with amplitude <IMG 
src="Reflection of Waves from Boundaries_files/xi1.gif"> in medium 1 encounters 
a boundary with medium 2, the amplitudes of the reflected and transmitted waves 
are determined by<BR>
<CENTER><IMG 
src="Reflection of Waves from Boundaries_files/Eq-1.jpg"></CENTER><BR>In the 
animations below, two strings of different densities are connected so that they 
have the same tension. The density of the thick string is 4 times that of the 
thin string. If the speed of waves on a string is related to density and tension 
by <BR>
<CENTER><IMG 
src="Reflection of Waves from Boundaries_files/Eq-2.jpg"></CENTER><BR>how do the 
wave speeds compare for the two strings? 
<H3>From high speed to low speed (low density to high density)</H3>
<TABLE>
  <TBODY>
  <TR>
    <TD><IMG src="Reflection of Waves from Boundaries_files/reflect1.gif" 
      width=300 align=left></TD>
    <TD vAlign=top>In this animation the incident wave is travelling from a 
      low density (high wave speed) region towards a high density (low wave 
      speed) region. 
      <P>
      <DL>
        <DD>=&gt; How do the amplitudes of the reflected and transmitted waves 
        compare to the amplitude of the incident wave?</I> 
        <DD>=&gt; How do the polarities of the reflected and transmitted waves 
        compare to the polarity of the incident wave?</I> 
        <DD>=&gt; How do the widths of the reflected and transmitted waves 
        compare to the width of the incident wave?</I> 
</DD></DL></TD></TR></TBODY></TABLE>
<H3>From low speed to high speed (high density to low density)</H3>
<TABLE>
  <TBODY>
  <TR>
    <TD><IMG src="Reflection of Waves from Boundaries_files/reflect2.gif" 
      width=300 align=left></TD>
    <TD vAlign=top>In this animation the incident wave is travelling from a 
      high density (low wave speed) region towards a low density (high wave 
      speed) region. 
      <P>
      <DL>
        <DD>=&gt; How do the amplitudes of the reflected and transmitted waves 
        compare to the amplitude of the incident wave?</I> 
        <DD>=&gt; How do the polarities of the reflected and transmitted waves 
        compare to the polarity of the incident wave?</I> 
        <DD>=&gt; How do the widths of the reflected and transmitted waves 
        compare to the width of the incident wave?</I> 
</DD></DL></TD></TR></TBODY></TABLE>
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